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Proof step 53 from proof infinite-append-prefixnextli-weak (previous step)

ii(0) = 0, ii(n0 + 1) = ii(n0) + # nextli0(lf(n0)), lf(n0 + 1) = nextli(lf(n0)), lf(0) = [], lf(n0) = (σ to ii(n0)).l, nextli(lf(n0)) = lf(n0) + nextli0(lf(n0)), (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ x. nextli(x) = x + nextli0(x), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ ⊦ # σ < ⌜ii(n0 + 1)⌝ , # nextli(lf(n0)) = # (σ to ii(n0 + 1)).l ∧ (∀ n. n < # nextli(lf(n0)) → nextli(lf(n0))[n] = (σ to ii(n0 + 1)).l[n])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 54:
ii(n0 + 1) = ii(n0) + # nextli0(lf(n0)), ii(0) = 0, lf(n0) = (σ to ii(n0)).l, lf(0) = [], lf(n0 + 1) = nextli(lf(n0)), nextli(lf(n0)) = lf(n0) + nextli0(lf(n0)), (σ from ii(n0)) to # nextli0(lf(n0)) = ⌜nextli0(lf(n0))⌝ , (finp(σ) ⊃ # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → nextli0(lf(m)) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # nextli0(lf(n)) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ (σ from ii(n)) to # nextli0(lf(n)) = ⌜nextli0(lf(n))⌝ , ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. lf(n + 1) = nextli(lf(n)), ∀ x. nextli(x) = x + nextli0(x) ⊦ # σ < ⌜ii(n0 + 1)⌝ , # nextli(lf(n0)) = ii(n0 + 1) ∧ (n1 < # nextli(lf(n0)) → nextli(lf(n0))[n1] = σ[n1])

Rule argument

Additional information

Used simplifier rules: m < n, ¬ # σ < ⌜n⌝ ⊦ (σ to n).l[m] = σ[m]; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ # x = 0 ↔ x = []; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)%¬ # σ < ⌜n⌝ ⊦ # (σ to n).l = n; (s)